Regardless of whether you use precomps or not, your main problem is with Collapse Transformations. In most cases you don't want to use it since it changes the AE render order. It causes transformations of the subcomp (transforms applied to the subcomp layer inside the main comp) to be applied to the layers inside the subcomp before effects and masks inside the subcomp.
For your type of project you can really only use Coll Trans if you don't use any masks or effects at all to either the layers inside the subcomp or the subcomp itself. (at least not without causing massive grief)
If you keep everything in one comp like the others have suggested you can use Cont Rast on the Illustrator layers so long as you realize any effects on these layers will render after transforms. This is reverse of how it normally is. It will also render masks after transformations which makes them hard to use.
I have done quite a few project of somewhat similar nature to what you're doing, and this is my method:
I use subcomps like you do. I make the subcomp about twice as big as it needs to be (or more). This means in the main comp I scale it down, and I can move it around and scale it up/down a bit while remaining sharp.
If you use Illustrator layers, scale the layers up so they fit this larger subcomp, and enable Cont Rast. This will make them sharp inside the subcomp. DON'T enable Coll Trans on the subcomp itself.
With this technique you can also add effects to the subcomp without running into trouble. You can also use adjustment layers and whatever else inside the subcomp.
For shadow, duplicate the subcomp layer inside the main comp, and use the second comp for shadow. You can then add tint effects, blur, and so on.
Or use 3D to cast shadows.
The general rule is: Don't use Collapse Transformations on subcomps unless you have a very good reason. If you must, there are severe limitations you have to be aware of.
Bit of a long explanation, but hope it helps.
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http://www.jonbagge.net
Jon Bagge - Editor - London, UK
Avid - FCP - After Effects